Trim the chicken with a paring knife and make several small slits all over each chicken thigh. This will enable the marinade to soak in a bit.
In a large saucepan whisk together the soy sauce, water, brown sugar, grated ginger, and garlic.
Place the chicken in a large ziplock bag and pour 3/4 cup of the soy sauce mixture over the chicken. Seal the bag and shake to coat the chicken. Refrigerate for at least 1 hour or up to 1 day.
Place the saucepan with the remaining soy sauce mixture over high heat until it begins to boil. Then, reduce to medium-high heat and cook, whisking occasionally, until the sauce is reduced to about 2/3 cup and it has thickened, about 5-10 minutes (NOTE: The sauce Teriyaki sauce will thicken even more as it cools).
Preheat your grill by heating all of the burners to high heat for 15 minutes, and then reduce the heat to medium-low. Scrape the cooking grates and oil them lightly with canola oil. Place the chicken skin-side up and cover the grill. Grill the chicken until the internal temperature of the meat reaches 120 degrees F, about 15-20 minutes.
Flip the chicken and cover the grill. Continue to cook the chicken until the skin is a nice deep golden color and the internal temperature reaches 160 degrees F, about another 15-20 minutes.
Flip the chicken over again so the pieces are skin-side up. Use about 1/3 cup of the teriyaki sauce to brush over the chicken and cover and grill until the glaze is nice an caramelized, about 5 minutes.
Move the chicken to a large plate or serving platter and let it rest for 5-10 minutes before serving. Serve with remaining Teriyaki sauce.